Survey of control performance in quantum information processing

Abstract

There is a rich variety of physics underlying the fundamental gating operations for quantum information processing (QIP). A key aspect of a QIP system is how noise may enter during quantum operations and how suppressing or correcting its effects can best be addressed. Quantum control techniques have been developed to specifically address this effort, although a detailed classification of the compatibility of controls schemes with noise sources found in common quantum systems has not yet been performed. This work numerically examines the performance of modern control methods for suppressing decoherence in the presence of noise forms found in viable quantum systems. The noise-averaged process matrix for controlled one-qubit and two-qubit operations are calculated across noise found in systems driven by Markovian open quantum dynamics. Rather than aiming to describe the absolute best control scheme for a given physical circumstance, this work serves instead to classify quantum control behavior across a large class of noise forms so that opportunities for improving QIP performance may be identified.

Appendix

1.1 Trotter–Suzuki and Solovay–Kitaev control

Both TS and SK methods utilize the identity that imperfect rotations by integer units of $\pi$ can accumulate and eventually cancel out the error through [17]

$$\begin{aligned} M(2k\pi ,\phi )&= \pm R(2k\pi \epsilon ,\phi ), \quad k = 1,2,... \end{aligned}$$

(45)

For TS control, sets of expansion coefficients $p_{jn}$ are chosen such that

$$\begin{aligned} R(\epsilon ,\phi )&= \varPi _{j,i} \exp (ip_{jn} A_i) + O(\epsilon ^{n+1}), \end{aligned}$$

(46)

for sets of non-commuting matrices $\{A_i\}$. With individual qubit rotation in the X-Y plane, it can be shown that a even-ordered correction sequences $R_{TS2k}$ can be constructed through the identities

$$\begin{aligned} \text {N2k}(\theta ,\phi )&= \hat{S}_k(\phi _k,2) M(\theta ,\phi ), \end{aligned}$$

(47)

$$\begin{aligned} \hat{S}_1(\phi _1,m)&= M(m\pi ,\phi _1)M(2m\pi ,-\phi _1)M(m\pi ,\phi _1), \end{aligned}$$

(48)

$$\begin{aligned} \hat{S}_j(\phi _j,m)&= \hat{S}_{j-1}(\phi _j,m)^{4j-1} \hat{S}_{j-1}(\phi _j,-2m) \hat{S}_{j-1}(\phi _j,m)^{4j-1}. \end{aligned}$$

(49)

The recursive formula for the sequence in Eq. (47) uses only imperfect rotations along special angles $\phi _j$, and relies on the initial mapping $\hat{S}_1$ and the concatenated formula in Eq. (49). The angles of rotation are chosen to satisfy the decomposition in (46) and are given by

$$\begin{aligned} \phi _j&= \cos ^{-1}\left[ -\frac{\theta }{8\pi f_j}\right] \end{aligned}$$

(50)

$$\begin{aligned} f_j&= \left( 2^{2j-1}-2\right) f_{j-1},\quad f_1 = 1 \end{aligned}$$

(51)

The identities in Eq. (47) and (49) alongside angle choices from Eq. (50) yield the second sequence N2 of Eq. (39). Due to the long length of higher-order sequences, only N2 is simulated in this work. For example, the fourth-order sequence is given by

$$\begin{aligned} N4&= \hat{S}_2(\phi _2,2)^4 \hat{S}_2(\phi _2,-4) \hat{S}_2(\phi _2,2)^4 M(\theta ,\phi ), \quad \phi _2 = \cos ^{-1} (-\theta /48\pi ), \end{aligned}$$

(52)

SK methods utilize sets of $2\pi$ rotations that rely on suppressing errors related to commutators of lower-order commutators [18]. The first-order sequence SK1 of Eq. (40) can be shown to cancel first-order errors to $M(\theta ,\phi )$ for rotation angle choice of $\phi _1 = \cos ^{-1}\left( -\frac{\theta }{4\pi }\right)$:

$$\begin{aligned} \hat{\text {SK}}_{1}M(\theta ,\phi )&= \left[ I+i\frac{\theta }{2}\left( X\cos (\phi )+Y\sin (\phi )\right) \epsilon \right] \nonumber \\&\qquad \times \left[ I - i \frac{\theta }{2} \left( X\cos (\phi ) + Y\sin (\phi ) \right) (\epsilon +1 ) \right] \nonumber \\&= I -i \frac{\theta }{2} \left( X\cos (\phi ) + Y\sin (\phi ) \right) + O(\epsilon ^2) \nonumber \\&= R(\theta ,\phi ) + O(\epsilon ^2) . \end{aligned}$$

(53)

The higher-order SK sequences concatenate lower-order sequences and cancel subsequently higher-order errors related to commutators of lower orders. This suppression relies on an identity from the proof of the Solovay–Kitaev theorem that gives a relation between commutators of the lower-order rotation operators,

$$\begin{aligned} \exp (-iA\epsilon ^j) \exp (-iB\epsilon ^k) \exp (iA\epsilon ^j) \exp (iB\epsilon ^k)&= \exp (-i[A,B]\epsilon ^{j+k} + O(\epsilon ^{j+k+1}). \end{aligned}$$

(54)

Equation (54) yields the second-order terms $SK_{2} (\phi _2)$ of Eq. (43), and leads to the full error-correction sequence SK2 of Eq. (42).

1.2 Optimal control

Unconstrained primitive control utilizes an optimization to locate controls beyond the amplitude constraints of each system. Here we calculate the necessary gradient $\delta \chi _{ {\mathcal {E}}}/ \delta c(t)$ for a given control c(t) . The gradient of J with respect to c(t) can first be expressed as

$$\begin{aligned} \frac{\delta J}{\delta c(t)}&= - \frac{1}{2N^2} \text {Re} \left( \text {Tr} \left[ \chi _W^{\dag } \frac{\delta \chi _{{\mathcal {E}}(T)}}{\delta c(t)} \right] \right) , \end{aligned}$$

(55)

where the explicit time dependence upon $\chi _{{\mathcal {E}}}$ has now been shown for clarity. Calculation of $\delta \chi _{{\mathcal {E}}}(T) / \delta c(t)$ can be performed by examining the time evolution of a linear disturbance to $\chi _{{\mathcal {E}}}(T)$. First note that analogous to the unitary evolution operator in a closed quantum system, the time evolution of $\chi _{{\mathcal {E}}}(t)$ follows a similar equation of motion:

$$\begin{aligned} \frac{\partial }{\partial t} \chi _{{\mathcal {E}}}(t)&= {\mathcal {L}}(t) \chi _{{\mathcal {E}}}(t). \end{aligned}$$

(56)

Inserting a variation $\delta \chi _{{\mathcal {E}}}(t)$ into Eq. (56) and keeping first-order terms yield an equation for evolution of dynamical disturbances to the process matrix as

$$\begin{aligned} \frac{\partial }{\partial t} \delta \chi _{{\mathcal {E}}}(t)&= {\mathcal {L}}(t) \delta \chi _{{\mathcal {E}}}(t) + \delta {\mathcal {L}}(t)\chi _{{\mathcal {E}}}(t) , \end{aligned}$$

(57)

where the $\delta L(t)$ corresponds to a variation of the control in the superoperator.

$$\begin{aligned} \delta {\mathcal {L}}(t)&= -i \left[ {\mathbb {I}}_{N} \otimes \delta H(t) - \delta H(t) \otimes {\mathbb {I}}_{N} \right] . \end{aligned}$$

(58)

For example, in the case of an X rotation in the RWA (i.e., Eq. (6), $\phi =0$, $\delta \omega (t) = 0$), this term is expressed as

$$\begin{aligned} \delta {\mathcal {L}}(t)&= -\frac{i \delta c(t)}{2} \left[ {\mathbb {I}}_{N} \otimes X - X \otimes {\mathbb {I}}_{N} \right] \end{aligned}$$

(59)

Equation (57) is an inhomogeneous differential equation and has a homogenous solution with Green’s function $G(t,t')$,

$$\begin{aligned} \frac{\partial }{\partial t} G(t,t')&= {\mathcal {L}}(t) G(t,t'). \end{aligned}$$

(60)

The above equation can be identified as the time evolution of the process matrix in Eq. (56), and $G(t,t')$ represents evolution of the process matrix from time point $t'$ to time t. Expressing this Green’s function in a separable form $G(t,t')$ relates it to the process matrix,

$$\begin{aligned} G(t,t')&= G(t,0)G(0,t') \nonumber \\&= G(t,0) G^{\dag }(t',0) \nonumber \\&= \chi _{{\mathcal {E}}}(t)\chi ^{\dag }_{{\mathcal {E}}}(t') . \end{aligned}$$

(61)

The inhomogeneous solution to $\delta \chi _{{\mathcal {E}}}(T)$ can now be expressed as

$$\begin{aligned} \delta \chi _{{\mathcal {E}}}(T)&= \int _0^T G(T,t) \delta {\mathcal {L}}(t) \chi _{{\mathcal {E}}}(t) \mathrm{d}t \nonumber \\&= \chi _{{\mathcal {E}}}(T) \int _0^T \chi ^{\dag }_{{\mathcal {E}}}(t) \delta {\mathcal {L}}(t) \chi _{{\mathcal {E}}}(t) \mathrm{d}t. \end{aligned}$$

(62)

The gradient $\delta \chi _{{\mathcal {E}}}(T) / \delta c(t)$ can then be identified from the total first-order variation in Eq. (62) as

$$\begin{aligned} \frac{\delta \chi _{{\mathcal {E}}}(T) }{\delta c(t)}&= \chi _{{\mathcal {E}}}(T) \chi ^{\dag }_{{\mathcal {E}}}(t) \frac{\partial {\mathcal {L}}(t)}{\partial c(t)} \chi _{{\mathcal {E}}}(t). \end{aligned}$$

(63)

and the entire gradient of J can then be explicitly written from Eqs. (63) and (55) as

$$\begin{aligned} \frac{\delta J}{\delta c(t)}&= - \frac{1}{2N^2} \text {Re} \left( \text {Tr} \left[ \chi _W^{\dag } \chi _{{\mathcal {E}}}(T) \left( \chi ^{\dag }_{{\mathcal {E}}}(t) \frac{\partial {\mathcal {L}}(t)}{\partial c(t)} \chi _{{\mathcal {E}}}(t) \right) \right] \right) , \end{aligned}$$

(64)

Optimal control pulses were found through the D-MORPH algorithm of Eqs. (34) and (35) to minimize J. As the quantum dot qubit system was excluded from optimal control, all systems were simulated under the RWA, in which controls were either a bias pulse in the Z direction, or the amplitudes for an on-resonance field in the XY plane. To explore the performance beyond primitive gate operations with constant amplitudes, a Gaussian temporal envelope amplitude was instead used for initial guesses $c_i(t)$. For rotation through angle $\theta$, the initial guess is expressed as

$$\begin{aligned} c(t)&= \frac{\theta }{T}\exp \left[ -\frac{8\pi }{T^2}(t-T/2)^2 \right] . \end{aligned}$$

(65)